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## Roots of Unity

A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

# Level 4

$\cos\left( \frac{\pi}{11}\right) - \cos\left( \frac{2\pi}{11} \right)+ \cos\left( \frac{3\pi}{11}\right) \\ - \cos\left( \frac{4\pi}{11} \right)+ \cos\left( \frac{5\pi}{11} \right)=\ ?$

Given $$z^2+z+1=0,$$ find the value of

$\left(z+\dfrac{1}{z} \right)^2+\left(z^2+\dfrac{1}{z^2}\right)^2+\left(z^3+\dfrac{1}{z^3} \right)^2+\cdots+\left(z^{21}+\dfrac{1}{z^{21}}\right)^2.$

$\large{\dfrac{31-2\delta_{1}}{1-2\delta_{1}} +\dfrac{31-2\delta_{2}}{1-2\delta_{2}}+\dfrac{31-2\delta_{3}}{1-2\delta_{3}}}$

If $$\large{1,\delta_{1},\delta_{2},\delta_{3}}$$ are distinct fourth roots of unity, then evaluate the expression above.

If $$w=e^{2\pi{i}/5}$$, find $\sum_{k=1}^{4}\frac{1}{1+w^k+w^{2k}}$

If for $$z \in \mathbb{C} \space ,$$ $$z+\dfrac{1}{z}=2 \cos 6°.$$ Then find the value of $\left( z^{1000}+\frac{1}{z^{1000}} \right).$

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